From the title of the thread. The PJ believed the dogs to be infallible. Thus the sold a pup idiom.
I'm afraid, John, that the PJ's belief is a myth. They ignored such dogs existed (did you know? I didn't), in France for instance there are 12 and they resolve about 15 cases a year.
The PJ trusted the NPIA and I think that was the right thing to do. If the UK dogs hadn't reacted at all, you could suspect the travel, the heat, the lack of cold country, but they reacted in determined situations and, had they not reacted it, that would have been also very revealing.
If they hadn't reacted in 5A, they wouldn't have been sent to the villa and the Scenic wouldn't have joined RM's cars which were the original target defined by Prof. Harrison.
Nobody would suspect that Madeleine died in the flat, though after so many years people would suspect she had some kind of fatal accident outside. The hypotheses would be implausible abduction from bed vs plausible wandering off/accidental death.